Exercise 30, § 6

Algebra, I, 155

Bourbaki marks this one as harder than the rest.

Let $p$ be a prime number, $r$ and $t$ integers $\geqslant 0$ , S a cyclic group of order $p^{r+t}$ and T the subgroup of S of order $p^t$ . (a) If S operates on a finite set E, show that

\operatorname{Card}(E) \equiv \operatorname{Card}(E^T) \pmod{p^{r+1}},

where

E^T

denotes the set of elements of E invariant under T. (b) Let

m, n \geqslant 0

. Show that

\binom{p^{r+t}m}{p^tm} \equiv \binom{p^rm}{n} \pmod{p^{r+1}}.

(Let M be a set with m elements and E the set of subsets of

S \times M

with

p^t n

elements. Define an operation of S on

S \times M

such that there exists a bijection of

E^T

onto the set of subsets of

(S/T) \times M

with n elements; apply (a).)

Solution

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